= Condensate compressibility in an optical lattice
{title2=$\kappa^{-1}=n(\partial\mu/\partial n)_s$}
Here $n$ is average <number density> and $\mu$ the ground-state <chemical potential>. This commonly used condensate response has units of inverse energy; thermodynamic <isothermal compressibility> is instead $\kappa/n$. In a weakly interacting repulsive condensate, the normalized one-cell ground state $f_s$, with average $|f_s|^2=1$, gives $\mu=E_0(s)+Un I(s)+\cdots$, $I(s)=d^{-1}\int_0^d|f_s|^4dx$. Localization raises $I$ and reduces $\kappa$ at fixed $n,U,d$. In deep harmonic wells $I\propto s^{1/4}$, hence $\kappa\propto s^{-1/4}$. This is a ground-band, repulsive-superfluid statement with the held variables specified, not a theorem for arbitrary signs of the interaction or arbitrary excited branches.
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